Annals of Applied Probability

Normal convergence of nonlocalised geometric functionals and shot-noise excursions

Raphaël Lachièze-Rey

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This article presents a complete second-order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results do not require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case, the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.

Article information

Ann. Appl. Probab., Volume 29, Number 5 (2019), 2613-2653.

Received: July 2018
Revised: October 2018
First available in Project Euclid: 18 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields 60F05: Central limit and other weak theorems

Poisson functionals shot-noise fields random excursions central limit theorem stabilisation Berry–Esseen bounds


Lachièze-Rey, Raphaël. Normal convergence of nonlocalised geometric functionals and shot-noise excursions. Ann. Appl. Probab. 29 (2019), no. 5, 2613--2653. doi:10.1214/18-AAP1445.

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